Kellerer, Albrecht M. (1983): On the Number of Clumps Resulting from the Overlap of Randomly Placed Figures in a Plane. In: Journal of applied probability, Vol. 20, Nr. 1: S. 126-135




When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.